demographic research volume 5, article 1, pages 1-22 … · 2010. 3. 9. · anastasia kostaki 1...

Demographic Research a free, expedited, online journal of peer-reviewed research and commentary in the population sciences published by the Max Planck Institute for Demographic Research Doberaner Strasse 114 · D-18057 Rostock · GERMANY www.demographic-research.org

DEMOGRAPHIC RESEARCH VOLUME 5, ARTICLE 1, PAGES 1-22 PUBLISHED 19 SEPTEMBER 2001 www.demographic-research.org/Volumes/Vol5/1/ DOI: 10.4054/DemRes.2001.5.1

Expanding an abridged life table

Anastasia Kostaki

Vangelis Panousis

© 2001 Max-Planck-Gesellschaft.

Table of Contents

1. Introduction 2

2. Expanding methodology 3

3. Comparisons 9

4. Concluding remarks 11

References 19

Demographic Research - Volume 5, Article 1

http://www.demographic-research.org 1

Expanding an abridged life table

Anastasia Kostaki 1

Vangelis Panousis 2

Abstract

A question of interest in the demographic and actuarial fields is the estimation of theage-specific mortality pattern when data are given in age groups. Data can be providedin such a form usually because of systematic fluctuations caused by age heaping. This isa phenomenon usual to vital registrations related to age misstatements, usuallypreferences of ages ending in multiples five. Several techniques for expanding anabridged life table to a complete one are proposed in the literature. Although many ofthese techniques are considered accurate and are more or less extensively used, theyhave never been simultaneously evaluated. This work provides a critical presentation,an evaluation and a comparison of the performance of these techniques. For thatpurpose, we consider empirical data sets for several populations with reliable analyticaldocumentation. Departing from the complete sets of qx-values, we form the abridgedones. Then we apply each one of the expanding techniques considered to theseabridged data sets and finally we compare the results with the corresponding completeempirical values.

1 Department of Statistics, Athens University of Economics and Business, 76 Patision St., Athens 10434,

Greece. Email: kostaki@ aueb.gr2 Department of Statistics, Athens University of Economics and Business, 76 Patision St., Athens 10434,

Greece. Email: [email protected]


2 http://www.demographic-research.org

1. Introduction

Mortality and fertility form the two general causes of the physical notion of a humanpopulation. Both causes are object of great attention and analysis by demographers. Ademographer constructs a life table in order to describe the survivorship of a populationas subjected to the risk of death. Such a table describes the effect of mortality. Anabridged life table presents the mortality pattern by age groups. The problem ofestimating a complete life table, when the data are provided in age intervals has beenextensively discussed in demographic, biostatistical, as well as actuarial literature. Themain reason for providing data in an abridged form are related to the phenomenon ofage heaping, caused by age misstatements in data registration. Another typical reason isthe unstable documentation of vital statistics based on insufficiently small samples. Themost typical age-misstatements is the preference of ages ending in multiples of five.Such misstatements cause the appearance of age heaps. An example of such agepreference in empirical data is illustrated below. The data are the observed number ofdeaths of the Greek female population in 1960.

The common approach to overcome this source of systematic errors adopted by thevarious statistical offices is to group the data in five-year age groups having themotivation that these over- and understatements which affect the age-specific data setswill efface each other into the five-year age group.

The Mortality of the 1960 Greek Female Population

age x

81777369656157534945

deat

hs d

x

2000

1800

1600

1400

1200

1000

800

600

400

200

0



However, for several reasons, there is a need for reliable estimations of mortalityrates specific by age. Several methods have been suggested in the literature forreproducing the age-specific mortality pattern. Techniques that utilize some generalinterpolation formula as King (1914), Beers (1944) have been extensively usedpreviously. An interpolation method that is still very actual is the six-term Lagrangeanformula to be applied to the existed in an abridged life table, lx values. This method ispresented in Abramowitz and Stegun (1965) and it forms the only expansion techniqueproposed in standard textbooks as Elandt-Johnson and Johnson (1980), and Namboodiri(1991). J.Pollard (1989) presents a relatively new one that requires five-year agegrouping. Another technique for expanding an abridged life table has developed byKostaki (1987) and also developed independently within the MORTPAK andMORTPAK-LITE software packages published by the United Nations (1988a, 1988b).For the application of this technique a parametric model of human mortality is required.In Kostaki (1987), as well as in the MORTPAK package the eight-parametric model ofHeligman and Pollard (1980) is used. Additionally, in Kostaki (1991) a final adjustmentto the results of this expanding technique is also developed. Some years later, Kostaki(2000) presented a non-parametric technique, in the sense that it does not require theuse of a parametric model. The new method relates the target abridged life table with anexisted complete one that is utilized as a standard. Interpolation via splines is a case ofan osculatory interpolation that has lately received great attention. Great literature referto splines, as tools for expanding an abridged life table (see e.g. Hsieh, 1991, McNeil,et. al., 1977, Wegman and Wright, 1983).

Although most of these techniques are considered accurate and are extensivelyused, their performance has never been compared and simultaneously evaluated. Thispaper provides an evaluation and a comparison of the performance of the varioustechniques. For that purpose, we consider empirical data sets for several populationswith reliable analytical documentation. Departing from the complete set of qx-values ofeach population, we form the corresponding abridged ones. We then apply each one ofthe expanding techniques considered to the abridged data sets and finally we comparethe resulting age-specific probabilities with the age-specific empirical values.

Next section is devoted to a presentation of the different techniques. Then inSection 3 we present and evaluate the results of our calculations while in the last sectionsome concluding remarks are provided.

2. Expanding methodology

Considering first the interpolation techniques, a conventional and extensively usedmethod for expanding an abridged life table (see e.g. Elandt-Johnson and Johnson,



1980) is the use of the six-term Lagrangean interpolation formula on the existing in the

abridged life table, xl -values. This formula expresses each non-tabulated value xl as a

linear combination of six particular polynomials in x, each of degree five,

∑ ∏

∏

=≠

≠

−

−=

6

1ii

ijji

ijj

)x(l)xx(

)xx()x(l (1)

where x1, x2, …, x6 are the tabular ages nearest to x.Let us now turn to spline interpolation. There is some enormous literature on

splines, most of it concerning their numerical analytic properties (see e.g. DeBoor,1978, Greville, 1969, Tyrtyshnikov, 1997), rather than their statistical properties (seee.g. Green, Silverman, 1995, McNeil, et.al., 1977, Härdle, 1992 or, Wahba, 1975).Special interest has been shown lately to such osculatory polynomials on the course ofDemography.

A spline, symbolized as S, is a function derived after joining a sequence ofpolynomial arcs. Suppose that bxxxa no =<<<= +11 ... , then a spline S of

degree k defined on the interval [a, b] with internal “ knots ” nxx ,...,1 is a function

such that, if ni ≤≤0 and 1+≤≤ ii xxx , then )()( xpxS i= where )(xpi is a

polynomial in x of degree not greater than k . The polynomial arcs meet at pivotalvalues are called knots. Splines are related to the roughness penalty approach of Greenand Silverman (1995). The roughness penalty approach for curve estimation is stated asfollows.

Given any twice - differentiable function S defined on [a,b], and a smoothingparameter ?, define the modified sum of squares,

∑ ∫ ′′λ+−==

n

1i

ba

22ii

functionpenalty roughness

dx))x(S( ))t(SY( )S(L 44 344 21

The penalized least squares estimator S is defined to be the minimizer of the functional)S(L over the class of all twice - differentiable functions S. This minimizer is the case

of a cubic spline. From )S(L , we can consider the following cases of a function



S:

→

∞→λ

. a to tend willS then, 0

. a to tend willS then , if

curve inginterpolat smooth

model regressionlinear

A simple representation of the interpolating spline of degree k can be written in thefollowing way.

∑∑=

+=

−+=n

i

kii

k

j

jj xxcxdxS

10

)( )(

where).0)(( otherwise zero and 0)( if , )( <−≥−−=− + iiii xxxxxxxx

The x’s into the formula are the data points where the polynomials meet (i.e. theknots) while d’s and c’s are parameters. They are all of number n+m+1. For k=3 onecan get the cubic spline interpolant. Two cases of the cubic spline are the most famous:

The Natural cubic spline

Among the set of all smooth curves S in ],[ baS interpolating the points ),( ii zx , the

one minimizing ∫ 2"S of L(S) is a natural cubic spline with knots ix . Furthermore

provided 2≥n there is exactly one such natural spline interpolant. This is introducedby the uniqueness theorem that says that supposing that 2≥n and nxx << ...1 ,

given any values nzzz ,...,, 21 , there is a unique natural cubic spline S with knots at

the points ix satisfying,

n,...1i for ,z)x(S ii ==

The Complete cubic spline

Hsieh (1991) considers the complete cubic spline. According to Hsieh, it appears to be amore accurate solution when applied to a life table construction problem. It has the



same definition of the natural one but with no natural conditions applied to the ends ofthe data. Hsieh (1991) gives further details for the end conditions.

McNeil, et.al. (1977), present the first analytic application of spline interpolationto demographic data. They present and adopt the cubic spline solution to a demographicinterpolation problem concerning fertility for Italian women of 1955. Such anapplication has to deal with rates that begin by age x=15, and not by the age zero whichis the case when we have to interpolate the xl - values of a life table.

Benjamin and Pollard (1980) present in detail spline smoothing and interpolationin mortality analysis. By adopting the natural cubic spline, they produce the graduatedmortality rates for a certain life table example. McCutcheon (1981), deals with certainaspects of splines at a greater length. Applications are concerned with the constructionof some U.K. National Life Tables. Details of that may also be found in the work ofMcCutcheon and Eilbeck (1977). An interesting survey article on splines in Statistics isthe one of Wegman, and Wright (1983), that attempts to synthesize a variety of workson splines in Statistics.

Later developments of the cubic spline mostly deal with fertility data. Nanjo(1986, 1987) describes the use of rational spline functions, which include cubic splineas a special case in obtaining data for single years of age for births given by five-yearage groups of mothers. Barkalov (1988) also describes the use of rational splines fordemographic data interpolation. Bergström, and Lamm (1989) apply cubic splineinterpolation in order to recover event histories. The method is used in order to adjustthe U.S. age at marriage schedules explaining a substantial part of the discrepancy inthe 1960 and 1970 censuses.

Hsieh (1991a,b) presents interesting applications of splines on Canadian data. Theauthor develops a set of new life table functions, which also includes in the formulationof what he calls complete cubic spline interpolation method.

Applications concerning splines in Demography also appear later in the literature,all of them concerning the smoothing spline case (see, e.g., Biller, and Färhmeir, 1997,Green, and Silverman, 1995, Renshaw 1995). Finally, Haberman & Rensaw (1999)present the more recent application of spline functions in the course of Demography bydeveloping a simple graphical tool for the comparison of two mortality experiences.Readers interested to study the subject of spline functions further, may consult, Greville(1969), DeBoor (1978), Härdle (1992), Green and Silverman (1995).

A relatively new procedure for expanding an abridged life table is the parametricone of Kostaki (1987, 1991). In that a parametric model which presents mortality as afunction of age is utilized. Kostaki (1987, 1991) has used the classical eight-parameterformula of Heligman and Pollard (1980) which presents the odds of mortality as aparametric function of age x, according to the formula:



qq

A D E x F GHx

x

x B xC

12

−= + − ++( ) exp( (ln( / )) ) (2)

Using the notation C for the parameters of the model (2) and the notation F(x; C) forthe right hand side of (2), we get

qF x C

F x Cx =+

=( ; )

( ; )1(3)

= G x C( ; )

say, and hence the relation

n x x ii

n

q q= − − +=

−

∏1 10

1

( )

implies the model

n xi

n

q G x i C= − − +=

−

∏1 10

1

( ( ; )

= n G x C( ; )

say, the later being an explicit but complicated function of C, x, and n.Thus given the values of nqx of the abridged life table one provides estimates of Cminimizing

(( ; )

)n

n xx

G x Cq∑ −1 2 (4)

where the summation is over all relevant values of x. Then inserting this C into (3)estimates of the one-year qx-values are produced.

An additional adjustment of the results of this expansion technique is presented inKostaki (1991). It entails the calculation of a constant K,



Kq

q

n x

x ii

n=−

− +=

−

∑ln( )

ln( ~ )

1

10

1

and the adjustment of the results ~q x through calculation of

~ ( ~ )′ = − −q qx xK1 1 .

This final adjustment produces results (~ )′q x fulfilling the property:

1 11

1

− − ′ ==

−

+∏i

n

x i n xq q( ~ ) .

Kostaki (2000) presents a simple non-parametric expanding technique. Accordingto that, the abridged nqx-values to be expanded are related to the one-year probabilitiesqx

(S) of another complete life table which is solely utilized as a standard.Under the assumption that the force of mortality, µ(x), underlying our abridged life

table is, in each n-year age interval [x, x+n), a constant multiple of the one underlyingthe standard life table in the same age interval, µ(S)(x), i.e.,

µ µ( ) * ( )( )x K xn xS= ,

the one-year probabilities qx+i , i= 0,1,..., n-1, in each n-year interval are equal to

1 1− − +( )( )q x iS Kn x (5)

where,

n xn x

x iS

i

nKq

q=

−

− +=

−

∑ln( )

ln( )( )

1

10

1 (6)

Thus, having the values of nqx of the abridged life table, as well as, the set of one-year ones, qx

(S) of the complete life table chosen to serve as a standard one (S), one



calculates the values of nKx using (6) and then inserting these values in (5) producesestimates for the one-year probabilities qx underlying the original abridged life table.

3. Comparisons

In order to evaluate and compare the performance of the various methods describedabove, we use empirical data from several populations and several time periods. Wehave the empirical qx - values which we initially abridge in wider age intervals for theages 0, 1-4, 5-9... e.t.c. Then we apply each one of the presented expanding methodsusing these abridged datasets. At a final step, we compare the resulting complete setswith the corresponding complete empirical ones.

For the purpose of the evaluation and comparison of the performance of thedifferent techniques used, we provide graphical representations of the results and wecalculate the values of two different criteria. These are the sum of squares of theabsolute and the relative deviations between the resulting and the empirical qx - values,

2)ˆ( xx

x qq∑ − (7)

and

2)1ˆ

( −∑x x

x

qq

(8)

The rationale behind the choice of criteria (7) and (8) is that they complete each othersince their values are indicative for the performance of a technique at the higherrespectively the earlier part of the total age interval.

The interpolating polynomial spline applied on the xl - values of the abridged life

table was obtained in S-plus (see Chambers and Hastie, 1992).In order to get estimates for the parameters of the Heligman&Pollard model we

use an iterative routine provided by the algorithm E04FDF of the NAG library. Thisalgorithm searches for the unconstrained minimum of the loss function (4).Additionally to the use of the minimization routine we supplied the subroutine LSFUN1in order to evaluate the loss function at each age x. In order to provide adequate startingvalues to the algorithm, the UNABR procedure of the MORTPAK package may beused. In some cases where the initial values were nor adequate the algorithm failed to



converge and stopped. In cases where the empirical evidence was very sparseMORTPAK did not suggested adequate initial values. In such cases we obtained initialparameter values using a stepwise estimation procedure proposed in Kostaki (1992)b. Inorder to obtain estimates for the standard errors of the parameters of HP8, we used theE04YCF routine also supplied by the NAG library.

In cases of data where the accident hump is not severe, the algorithm providesparameter estimates associated with very high standard errors. In such cases, the modelbecomes overparametrised, since its second term tries to estimate a hump of themortality curve that is almost nonexistent. Several authors e.g. Rogers (1986), Forfarand Smith (1987), Congdon (1993), Lytrokapi (1998), or Karlis and Kostaki (2000)refer to this problem. In such a case the resulting set parameter values loss theirinterpretability. This problem might be overcome if the second term of (2) is cut, i.e. byusing the model,

xB)(X

x

x GHApq C

+= + (8)

We investigated this idea using data of Norwegian females, where the accidenthump was almost disappeared but the fit obtained was not as good as when using thecomplete model. An illustration of the results is presented in Figure 3. It seems that thesecond term is always required to be incorporated in the model. Data reveal that asmall shift of the curve between the ages of 10 to 30 always exists. That cannot beassessed by only the two terms of the model. Anyway the problem in the performanceof the model (8) is similar to the one exhibited in the performance of (2) when fitted todata sets with intense accident hump. In the later case the use of the nine-parameterversion of Heligman-Pollard formula, proposed by Kostaki(1992) is recommended.Alternatively one can consider the model without the second term and estimate the

remaining terms separately, i.e. to estimate CB)(X

x

x Apq += , for the ages x ≤ x1 where

x1 is near 10, then estimate the other term of the model x

x

x GHpq

= , for the ages x ≥

x2 where x2 being near age 30 and then fill the gaps by performing spline interpolationto the whole-abridged data. In that way we may get more accurate estimates for theparameters concerning the childhood and the later adult ages. Of course in such a casewe cannot make statistical inference for the early adult ages.

The relational technique applied to each one of the abridged data sets consideredusing as standard the corresponding complete ones of the other sex.



Table 1 and 2 provide the values of (6) and (7) calculated using different data setsfor the common age interval of all methods considered. Figures 1-2 provide illustrationsof the performance of the various methods applied to these data.

We observe that splines had a good performance especially in the later part of themortality curve, the complete spline being more accurate than the natural one. Apeculiarity in the resulting complete sets of qx values is only exhibited at the childhoodages. This peculiarity was much more obvious for the Lagrange interpolation. Howeverthe later produce very accurate results for the adult ages.

The parametric method has a very good performance for the total age intervalwhich becomes even better after the adjustment, the results being closest to theempirical values though not smooth any more. When we try to avoid the second term ofHeligman-Pollard formula in order to achieve stability in parameters, in the case of anearly disappeared accident hump, the reduced model proved less accurate to reproducethe mortality pattern. Finally, we observe that the relational method of Kostaki (2000)though it’s simplicity produced accurate results.

4. Concluding remarks

In this paper, we reviewed the methods that one can adopt when faced with the problemof expanding an abridged life table to an analytical one. Summarizing, we distinguishedthese methods in: Firstly, piecewise polynomial interpolation techniques. Here wereviewed interpolation via splines and the Lagrangean interpolation, which forms amuch simpler technique when compared to the spline interpolation. Secondly methodsthat require the use of a parametric model. The Heligman-Pollard model of eightparameters was utilized here. However, as mentioned above the choice of theparametric model is open. (Kostaki and Lanke, 2000 have used the classical Gombertzformula). Finally, we reviewed a non-parametric relational technique of Kostaki (2000).

In order to evaluate the various methods we set out a set of basic properties.1) The ability of the expanding technique to accurately reproduce the age-

specific mortality pattern for the total age interval considered.All the techniques considered proved accurate to produce the age specific mortalitypattern. However the application of the Lagrangean formula is limited to the ages lessthan 75 while literature proposes other complementary methods for the ages greaterthan 75 (see Elandt-Jonshon and Johnson, 1980). Additionally, the shape of the age-specific mortality pattern resulting by the interpolation techniques is deformed at thelater childhood ages. The interpolated values exhibit a systematic fluctuation tracked atthe ages about 10=x to 15. A sudden drop at the birth ages mortality is also added tothe previous observation. This is more evident in the Lagrangean interpolation.



2) The ability of a method to obtain smooth results.The parametric method before the adjustment is the only method that provides smoothresults while its results after the adjustment are closest to the empirical ones. A splineinterpolant will also provide a roughly smooth mortality curve, a property distinguisheda spline from the other piecewise interpolation techniques. The Langrangeaninterpolation and the non-parametric relational technique will provide a series ofprobabilities that are not smooth. The later can produce smoother results if the standardtable utilized is a graduated one.

3) An important property for an expanding technique is to get again the inputabridged life table after abridging its results.This property is exhibited by the results of the parametric technique after the additionaladjustment as well as by the results of the relational technique.

4) The simplicity of a methodThe parametric technique though produce the best results in comparisons to the others,it requires advanced software. Advanced software is also required in order to constructthe piecewise interpolants in the case of spline interpolation. The Lagrangeaninterpolation requires much easier calculations, while the non-parametric technique isthe easiest one that can be performed even with the simplest calculator.

Summarizing we conclude that the parametric model solution before theadjustment offers a smooth curve, which extends to the whole age interval. In additionit provides flexible results in the sense that they can be represented in a set of parametervalues and this facilitates comparisons through time and space. The non-parametrictechniques lack the interpretability part that one can obtain from the parameters in amodel. Nevertheless, they provide accurate tools, most of them being much more easilyapplied than the parametric technique. A disadvantage of the interpolation techniques incontrast to the parametric and the relational ones is that their results if abridged they arenot equal to the original ones. Moreover, in contrast to the parametric and the relationaltechniques again, they do not take advantage to the fact that the function to beinterpolated is of a special type, viz. A survivor function. Therefore in some cases theydo not reproduce the typical shape of the age-specific mortality pattern.



Table 1: Values of the sum of squares of the absolute deviations between theresulting and the empirical qx -values , multiplied by 103.

Expanding Method

HP8NaturalSpline

CompleteSpline Lagrange

RelationalTechnique

HP8Adjustment

Life TableItaly females1990-91

0.01325 0.000805 0.000633 0.00073 0.00204 0.00100

Italy males1990-91

0.02574 0.00129 0.00133 0.00190 0.0438 0.00675

Sweden females1991-95

0.0111 0.00328 0.00515 0.00037 0.00101 0.00082

Sweden males1991-95

0.0040 0.0113 0.012 0.00083 0.0033 0.00089

New Zealandfemales 1980-82

0.0068 0.00035 0.00027 0.0004 0.00183 0.00106

New Zealandmales 1980-82

0.0226 0.00158 0.00163 0.0021 0.0054 0.00273

Norway females1951-55

0.0005 0.00093 0.0007 0.00057 0.00069 0.0005

Norway males1951-55

0.0010 0.0009 0.00076 0.00093 0.0013 0.0006

Table 2: Values of the sum of squares of the relative deviations between theresulting and the empirical qx -values.

Expanding Method

HP8NaturalSpline

CompleteSpline Lagrange

RelationalTechnique

HP8Adjustment

Life TableItaly females 1990-91

0.317 0.701 0.453 0.786 0.360 0.0650

Italy males 1990-91 0.205 0.447 0.327 1.231 0.301 0.0582Sweden females1991-95

1.061 1.065 0.925 1.172 1.235 0.961

Sweden males1991-95

1.032 0.762 0.718 2.226 0.271 0.800

New Zealandfemales 1980-82

0.356 0.668 0.378 0.750 0.356 0.0924

New Zealandmales 1980-82

0.283 1.014 0.808 2.097 0.392 0.218

Norway females1951-55

0.239 0.378 0.241 0.515 0.423 0.155

Norway males1951-55

0.255 0.292 0.232 0.616 0.589 0.161



Figure 1: Empirical xq – values (points) and estimations by (1) the HP8 formula(solid line), (2) the complete cubic spline interpolation (solid line), (3) thenatural cubic spline interpolation (solid line), (4) the Lagrangeaninterpolation (solid line), (5) the non-parametric relational technique(solid line) and (6) the HP8 adjustment, for Italian males 1990-91.

(1)-HP8

(2)- Complete Spline

age x

1009080706050403020100

ln(q

x*10

0000

)

10

9

8

7

6

5

4

3

2

1

0

Italy 1990-91

age x

1009080706050403020100

ln(q

x*10

0000

)

10

9

8

7

6

5

4

3

2

1

0 Italy 1990-91



(3)-Natural Spline (4) Lagrangean interpolation

(5) Relational technique (6) HP8 adjustment

age x

80706050403020100

ln(q

x*10

0000

)

9

8

7

6

5

4

3

2

1

0

Italy 1990-91

age x

1009080706050403020100

ln(q

x*10

0000

)

10

9

8

7

6

5

4

3

2

1

0 Italy 1990-91

age x

1009080706050403020100

ln(q

x*10

0000

)

10

9

8

7

6

5

4

3

2

1

0

Italy 1990-91

age x

1009080706050403020100

ln(q

x*10

0000

)

10

9

8

7

6

5

4

3

2

1

0

Italy 1990-91

Fitted



Figure 2: Empirical xq – values (points) and estimations by (1) the HP8 formula(solid line), (2) the complete cubic spline interpolation (solid line), (3) thenatural cubic spline interpolation (solid line), (4) the Lagrangeaninterpolation (solid line), (5) the non-parametric relational technique(solid line) and (6) the HP8 Adjustment, for Norwegian females 1951-55.

(1)- HP8

(2)- Complete Spline

age x

80706050403020100

ln(q

x*10

0000

)

9

8

7

6

5

4

3

2

1

0 Norway 1951-55

age x

80706050403020100

ln(q

x*10

0000

)

9

8

7

6

5

4

3

2

1

0 Norway 1951-55



(3)- Natural Spline (4) Lagrangean interpolation

(5) Relational technique 6)- HP8 Adjustment

age x

80706050403020100

ln(q

x*10

0000

)

9

8

7

6

5

4

3

2

1

0 Norway 1951-55

age x

6050403020100

ln(q

x*10

0000

)

8

7

6

5

4

3

2

1

0 Norway 1951-55

age x

6050403020100

ln(q

x*10

0000

)

8

7

6

5

4

3

2

1

0 Norway 1951-55

age x

80706050403020100

ln(q

x*10

0000

0

9

8

7

6

5

4

3

2

1

0 Norway 1951-55



Figure 3: Empirical xq – values (points) and estimations by the reduced HP8formula (solid line), for Norwegian females 1951-55.

age x

80706050403020100

ln(q

x*10

0000

0

9

8

7

6

5

4

3

2

1

0 Norway 1951-55



References

Abramowitz, M, I. A. (1965). Handbook of Mathematical Functions with formulas,Graphs, and Mathematical Tables, National Bureau of Standards, AppliedMathematics Series 55, 4th printing. Washington, D.C.: U.S. GovernmentPrinting Office.

Barkalov, N. B. (1988). Interpolation of demographic data using rational splinefunctions. Demograficheskie Issledovaniya, 44-64. (In Rus.)

Beers, S.H. (1944). Six - term formulae for Routine Actuarial Interpolation, R.A.I.A.,33(2), 245-260.

Benjamin, B. and Pollard, J.H. (1980). The Analysis of Mortality and other ActuarialStatistics, second edition, Heinemann, London.

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